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<title>Beginnings of Calculus</title>
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<H1>Beginnings of Calculus</H1>

<a href="http://www.maths.tcd.ie/pub/HistMath/People/RBallHist.html">
<i>Short Account of the History of Mathematics</i>
</a>
by W. W. Rouse Ball (4th Edition, 1908)


<H2>Cavalieri </H2>

<p/>
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Cavalieri.jpeg"
height=
"500" /><br/>
<img src="http://upload.wikimedia.org/wikipedia/commons/b/b7/Cavalieri%27s_principle.jpg"
height=
"500" />

<p/>

Biography in Dictionary of Scientific Biography
<br/>
<a href="http://www.britannica.com/eb/article-9021898">Encyclopaedia Britannica<a/>
(also: 
<a href="http://search.eb.com.ezproxy.csusm.edu/eb/article-9021898">Academic Edition<a/>,
requires campus password)
<br/>
Crater Cavalierius

<p/>
<a href="http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Cavalieri.html">
MACTUTOR<a/>. 
Bonaventura Cavalieri
<br/> Full name:
Bonaventura Francesco Cavalieri
<br/>
Born: 1598 in Milan, Duchy of Milan, Habsburg Empire (now Italy)
<br/>
Died: 30 Nov 1647 in Bologna, Papal States (now Italy)
<br>
Summary: 
Italian mathematician. Developed a method of indivisibles 
which became a factor in the development of the integral calculus.

<p/>
<a href="http://www.britannica.com/eb/article-9021898">
BRITANNICA<a/>. 
Cavalieri, (Francesco) Bonaventura
<br/>
Born: 1598, Milan [Italy]
<br/>
Died: Nov. 30, 1647, Bologna, Papal States
<br/>
Summary: Italian mathematician.
Made developments in geometry that were precursors to integral calculus.

<p/>
<a href="http://galileo.rice.edu/Catalog/NewFiles/cavaleri.html">
The Galileo Project<a/>

<p/>
<a href="http://www.maths.tcd.ie/pub/HistMath/People/Cavalieri/RouseBall/RB_Cavalieri.html">
Biography:</a> 
Bonaventura Cavalieri (1598 - 1647)<br/>
from <i>Short Account of the History of Mathematics</i>
by W. W. Rouse Ball (4th Edition, 1908)

<p/>
<a href="http://en.wikipedia.org/wiki/Bonaventura_Cavalieri">
WIKI<a/>.
Bonaventura Cavalieri
<br/>
Full name: Bonaventura Francesco Cavalieri.
Latin, Cavalerius (1598–November 30, 1647) 
<br/>
Summary: 
Italian mathematician.
Known for Cavalieri's principle: volumes of two objects are equal if the areas of their corresponding cross-sections are in all cases equal. 
Developed a "method of the indivisibles" to determine areas and volumes.
A significant step on the way to modern infinitesimal calculus.
<br/>
<i>Geometria indivisibilibus</i> (1635)
<br/>
<i>Trigonometria plana</i> (1636)
<br/>
<i>Exercitationes Geometricae</i> (1647)


<p/>
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Cavalieri.jpeg"
height=
"100" />
<img src="http://upload.wikimedia.org/wikipedia/commons/b/b7/Cavalieri%27s_principle.jpg"
height=
"100" />
<img src="http://upload.wikimedia.org/wikipedia/nl/1/10/Cavalieri.jpg"
height=
"100" />


<H2>Newton </H2>

<img src="http://upload.wikimedia.org/wikipedia/commons/3/39/GodfreyKneller-IsaacNewton-1689.jpg"
height=
"500" />
<img src=" http://www.astro.wisc.edu/~mab/education/astro103/lectures/figures/chap1/newton.jpg"
height=
"500" />
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<img src="http://upload.wikimedia.org/wikipedia/commons/c/cc/NewtonsTelescopeReplica.jpg"
height=
"550" />
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<img src="http://upload.wikimedia.org/wikipedia/commons/4/41/NewtonsPrincipia.jpg"
height=
"600" />
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<img src="http://upload.wikimedia.org/wikipedia/commons/d/d0/Principia_Page_1726.jpg"
height=
"700" />
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<img src="http://www.huntington.org/LibraryDiv/Newton/Optics.jpg" height="400"/>
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<img src="http://upload.wikimedia.org/wikipedia/commons/b/ba/Isaac_Newton_grave_in_Westminster_Abbey.jpg"
height=
"600" />
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<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_10.jpeg"
height=
"500" />
<br/>
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_23.jpeg"
height=
"220" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_14.jpeg"
height=
"220" />
<br/>
<img src="http://upload.wikimedia.org/wikipedia/en/0/0e/Newton-WilliamBlake.jpg"
height=
"500" />


<p/>
Biography in Dictionary of Scientific Biography
<br/>
<a href="http://www.britannica.com/eb/article-9108764">Encyclopaedia Britannica<a/>
(also: 
<a href="http://search.eb.com.ezproxy.csusm.edu/eb/article-9108764">Academic Edition<a/>,
requires campus password)
<br/>
Crater Newton
<br/>
Paris street name: Rue Newton (16th Arrondissement)
<br/
Follow of the Royal Society (elected 1672)

<p/>
<a href="http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Newton.html">
MACTUTOR<a/>. 
Sir Isaac Newton
<br/>
Born: 4 Jan 1643 in Woolsthorpe, Lincolnshire, England
<br/>
Died: 31 March 1727 in London, England
<br>
Summary: 
Greatest English mathematician of his generation.
Laid the foundation for differential and integral calculus. 
Work on optics and gravitation make him one of the greatest scientists the world has known.

<p/>
<a href="http://www.britannica.com/eb/article-9108764">
BRITANNICA<a/>. 
Newton, Sir Isaac
<br/>
Born: December 25, 1642 [January 4, 1643, New Style], Woolsthorpe, Lincolnshire, England
<br/>
Died: March 20 [March 31], 1727, London
<br/>
Summary:
English physicist and mathematician.
Culminating figure of the scientific revolution of the 17th century. 
Optics: discovered the composition of white light.
Mechanics: 3 laws of motion, 
law of universal gravitation. 
Mathematics: discoverer of calculus. 
Wrote: <i>Philosophiae Naturalis Principia Mathematica</i> (1687),
one of the most important single works in the history of modern science.

<p/>
<a href="http://plato.stanford.edu/entries/newton-stm/">
STANFORD 1<a/> (Newton's Views on Space, Time, and Motion)
<br/>
<a href="http://plato.stanford.edu/entries/newton-philosophy/">
STANFORD 2<a/> (Newton's Philosophy)
<br/>
<a href="http://www.newtonproject.ic.ac.uk/">
The Newton Project</a>
<br/>
<a href="http://galileo.rice.edu/Catalog/NewFiles/newton.html">
The Galileo Project<a/>
<br/>
<a href="http://www.maths.tcd.ie/pub/HistMath/People/Newton/RouseBall/RB_Newton.html">
Biography:</a> 
Sir Isaac Newton (1642 - 1727)<br/>
from <i>Short Account of the History of Mathematics</i>
by W. W. Rouse Ball (4th Edition, 1908)

<p/>
<a href="http://en.wikipedia.org/wiki/Isaac_Newton">
WIKI<a/>.
Sir Isaac Newton (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727] 
<br/>
Summary: 
English physicist, mathematician, astronomer, alchemist, and natural philosopher.
Regarded by many as the greatest figure in the history of science.
<i>Philosophiae Naturalis Principia Mathematica</i> (1687)
describes universal gravitation and the three laws of motion.
Derives Kepler's laws
Enunciated conservation of momentum and angular momentum. 
Inventor of the reflecting telescope.
Discovered the spectrum when white light passes through a prism.
Argued that light is composed of particles. 
Formulated an empirical law of cooling, studied the speed of sound, and proposed a theory of the origin of stars. 
Shares the credit with Gottfried Leibniz for the development of calculus. 
The generalized binomial theorem, "Newton's method" for approximating the zeroes,
and contributed to the study of power series.
Lagrange often said that Newton was the greatest genius who ever lived.
English poet Alexander Pope: "Nature and nature's laws lay hid in night;
God said "Let Newton be" and all was light".


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<img src="http://upload.wikimedia.org/wikipedia/commons/3/39/GodfreyKneller-IsaacNewton-1689.jpg"
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"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton.jpeg"
height=
"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_3.jpeg"
height=
"100" /><img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_2.jpeg"
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"100" />
<img src="http://upload.wikimedia.org/wikipedia/commons/3/39/GodfreyKneller-IsaacNewton-1689.jpg"
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"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_4.jpeg"
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"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_5.jpeg"
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"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_7.jpeg"
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<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_17.jpeg"
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"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_25.jpeg"
height=
"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_22.jpeg"
height=
"100" />
<img src="http://www.math.fu-berlin.de/rd/ag/isaac/newton/newtn3_f.jpg"
height=
"100" />
<img src=" http://www.astro.wisc.edu/~mab/education/astro103/lectures/figures/chap1/newton.jpg"
height=
"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_26.jpeg"
height=
"100" />


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<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_27.jpeg"
height=
"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_15.jpeg"
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"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_10.jpeg"
height=
"100" />
<img src="http://upload.wikimedia.org/wikipedia/commons/4/43/Bolton-newton.jpg"
height=
"100" />


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<img src="http://jeff560.tripod.com/newton4.jpg"
height=
"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_23.jpeg"
height=
"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_14.jpeg"
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"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_18.jpeg"
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"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_19.jpeg"
height=
"100" />
<img src="http://upload.wikimedia.org/wikipedia/he/1/19/Newton.jpg"
height=
"100" />


<p/>
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_2.jpeg"
height=
"100" />
<img src="http://upload.wikimedia.org/wikipedia/commons/3/39/GodfreyKneller-IsaacNewton-1689.jpg"
height=
"100" />
<img src="http://upload.wikimedia.org/wikipedia/commons/4/44/SS-newton.jpg"
height=
"100" />
From a portrait by Godfrey Kneller in 1689

<p/>
<img src=" http://www.astro.wisc.edu/~mab/education/astro103/lectures/figures/chap1/newton.jpg"
height=
"100" />
From a portrait (1702) by Kneller

<p/>
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_25.jpeg"
height=
"100" />
From a portrait (1712) by James Thornhill.

<p/>
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_4.jpeg"
height=
"100" />
<img src="http://www.math.fu-berlin.de/rd/ag/isaac/newton/newtn3_f.jpg"
height=
"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_5.jpeg"
height=
"100" />
From a portrait (1725) by John Vanderbank

<p/>
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton.jpeg"
height=
"100" />
From a portrait (1726) by Enoch Seeman

<p/>
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_10.jpeg"
height=
"100" />
This medallion was issued by the Royal Mint
in 1727 when Newton died. 

<p/>
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_8.jpeg"
height=
"100" />
<img src="http://upload.wikimedia.org/wikipedia/en/0/0e/Newton-WilliamBlake.jpg"
height=
"100" />
Newton portrayed as a divine geometer. (William Blake 1795).


<p/>
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Newton_9.jpeg"
height=
"100" />
Artemis holding Newton's portrait (by William Stukeley)

<p/>
<img src="http://upload.wikimedia.org/wikipedia/commons/c/cc/NewtonsTelescopeReplica.jpg"
height=
"100" />
Replica of Newton's reflector telescope (6 inch diameter mirror)

<p/>
<img src="http://upload.wikimedia.org/wikipedia/commons/4/41/NewtonsPrincipia.jpg"
height=
"100" />
<i>Philosophiae Naturalis Principia Mathematica</i>.
Newton's own copy of the first edition (with Newton's handwritten corrections for the
second edition).

<p/>
<img src="http://upload.wikimedia.org/wikipedia/commons/d/d0/Principia_Page_1726.jpg"
height=
"100" />

<p/>
<img src="http://www.huntington.org/LibraryDiv/Newton/Optics.jpg" height="100"/>
<img src="http://upload.wikimedia.org/wikipedia/commons/5/50/461px-Newton-opticks.jpg"
 height="100"/>
<img src="http://www.huntington.org/LibraryDiv/Newton/Storm.jpg" height="100"/>
From his <i>Optics</i>.

<p/>
<img src="http://upload.wikimedia.org/wikipedia/commons/b/ba/Isaac_Newton_grave_in_Westminster_Abbey.jpg"
height=
"100" />
Newton is buried in Westminster Abbey.


<H2>Leibniz </H2>


<p/>
<img src="http://upload.wikimedia.org/wikipedia/commons/6/6a/Gottfried_Wilhelm_von_Leibniz.jpg"
height=
"500" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Leibniz.jpeg"
height=
"500" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Leibniz_2.jpeg"
height=
"500" />
<img src="http://upload.wikimedia.org/wikipedia/eo/1/1c/Leibniz.jpg"
height=
"500" />


<br/>

<img src="http://www-gap.dcs.st-and.ac.uk/~history/Bookpages/Leibniz_machine.jpeg"
height=
"350" />
<br/>
<img src="http://upload.wikimedia.org/wikipedia/commons/9/92/Leibnitzrechenmaschine.jpg"
height=
"500" />

<br/>
<img src="http://upload.wikimedia.org/wikipedia/commons/1/1a/Leibniz-Denkmal.jpeg"
height=
"600" />
<img src="http://upload.wikimedia.org/wikipedia/commons/2/28/G%C3%B6ttingen-Auditorium-Leibniz.01.jpg"
height=
"600" /> 


<p/>

Biography in Dictionary of Scientific Biography
<br/>
<a href="http://www.britannica.com/eb/article-9047669">Encyclopaedia Britannica<a/>
(also: 
<a href="http://search.eb.com.ezproxy.csusm.edu/eb/article-9047669">Academic Edition<a/>,
requires campus password)
<br/>
Crater Leibnitz
<br/>
Paris street name: Rue Leibnitz and Square Leibnitz (18th Arrondissement)
<br/
Follow of the Royal Society (elected 1673)


<p/>
<a href="http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Leibniz.html">
MACTUTOR<a/>. 
Gottfried Leibniz
<br/>
Full name: Gottfried Wilhelm von Leibniz
<br/>
Born: 1 July 1646 in Leipzig, Saxony (now Germany)
<br/>
Died: 14 Nov 1716 in Hannover, Hanover (now Germany)
<br>
Summary: 
German mathematician.
Developed the present day notation for the differential and integral calculus.
Didn't think of the derivative as a limit. 
His philosophy also important.
Invented an early calculating machine.

<p/>
<a href="http://www.britannica.com/eb/article-9047669">
BRITANNICA<a/>. 
Gottfried Wilhelm Leibniz
<br/>
Born: July 1 [June 21, old style], 1646
<br/>
Died: November 14, 1716, Hannover, Hanover
<br/>
Summary:
German philosopher, mathematician, political adviser.
Important both as a metaphysician and as a logician.
Distinguished also for his independent invention of calculus.

<p/>
<a href="http://plato.stanford.edu/entries/leibniz-causation/">
STANFORD 1<a/> (Leibniz on Causation)
<br/>
<a href="http://plato.stanford.edu/entries/leibniz-mind/">
STANFORD 2<a/> (Leibniz's Philosophy of Mind)
<br/>
<a href="http://plato.stanford.edu/entries/leibniz-ethics/">
STANFORD 3<a/> (Leibniz's Ethics)
<br/>
<a href="http://plato.stanford.edu/entries/leibniz-evil/">
STANFORD 4<a/> (Leibniz on the Problem of Evil)
<br/>
<a href="http://galileo.rice.edu/Catalog/NewFiles/leibniz.html">
The Galileo Project<a/>
<br/>
<a href="http://www.maths.tcd.ie/pub/HistMath/People/Leibniz/RouseBall/RB_Leibnitz.html">
Biography:</a> 
Gottfried Wilhelm Leibnitz (1646 - 1716)<br/>
from <i>Short Account of the History of Mathematics</i>
by W. W. Rouse Ball (4th Edition, 1908)


<br/>
<a href="http://philosophy2.ucsd.edu/~rutherford/Leibniz/"/>
Texts and Translations</a> <br/>
<a href="http://www.leibniz-translations.com"/>
Translations</a> <br/>


<p/>
<a href="http://en.wikipedia.org/wiki/Leibniz">
WIKI<a/>.
Gottfried Leibniz
(July 1 (June 21 Old Style) 1646 – November 14, 1716) <br/>
Full name:
Gottfried Wilhelm Leibniz (also Leibnitz or von Leibniz)
<br/>
Summary: 
German polymath (wrote mostly in French and Latin).
Educated in law and philosophy.
Played a major role in the European politics and diplomacy of his day. 
Invented calculus independently of Newton.
Notation in general use since. 
Invented binary.
In philosophy: remembered for optimism
(best possible universe)
Contributions to physics and technology.
Anticipated notions in biology, medicine, geology, probability theory, psychology, and information science. 
Wrote on politics, law, ethics, theology, history, and philology, even occasional verse. 


<p/>
<img src="http://upload.wikimedia.org/wikipedia/commons/6/6a/Gottfried_Wilhelm_von_Leibniz.jpg"
height=
"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Leibniz.jpeg"
height=
"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Leibniz_2.jpeg"
height=
"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Leibniz_3.jpeg"
height=
"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Leibniz_4.jpeg"
height=
"100" />
<img src="http://upload.wikimedia.org/wikipedia/eo/1/1c/Leibniz.jpg"
height=
"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Leibniz_6.jpeg"
height=
"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Leibniz_8.jpeg"
height=
"100" />

<p/>
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Leibniz_5.jpeg"
height=
"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Leibniz_7.jpeg"
height=
"100" />
<img src="http://www-gap.dcs.st-and.ac.uk/~history/BigPictures/Leibniz_9.jpeg"
height=
"100" />

<p/>

<img src="http://www-gap.dcs.st-and.ac.uk/~history/Bookpages/Leibniz_machine.jpeg"
height=
"100" />

<img src="http://upload.wikimedia.org/wikipedia/commons/9/92/Leibnitzrechenmaschine.jpg"
height=
"100" />

<p/>
<img src="http://upload.wikimedia.org/wikipedia/commons/1/1a/Leibniz-Denkmal.jpeg"
height=
"100" />
(in Leipzig)
<p/>

<img src="http://upload.wikimedia.org/wikipedia/commons/2/28/G%C3%B6ttingen-Auditorium-Leibniz.01.jpg"
height=
"100" /> (in Göttingen)

<p/>


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